Local Projective Model Structures on Simplicial Presheaves
Local Projective Model Structures on Simplicial Presheaves
复制标题
单纯预滑轮上的局部射影模型结构
DOI：
10.1023/a:1013302313123
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
Benjamin A. Blander
中科院分区：
文献类型：
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作者：
Benjamin A. Blander
我们给出了在一些本质上很小的格罗滕迪克点T上的简单预轴类的模型结构。当T是Nisnevich点时，它专一于一个适当的简单模型类别，具有与[MV]中相同的弱等效，但颤振更少，因此颤振更多。这使得[V2]的比较定理的证明更简单，它不使用∆闭类。本说明的目的是在一些本质上很小的格罗滕迪克站点T上介绍简单预轴和简单轴类的不同模型结构，并给出这些简化模型类别的一些应用。特别地，我们证明了稳定同伦范畴SH（(Sm/k)Nis, a1）和SH（(Sch/k)cdh, a1）是等价的。这个结果最初是由Voevodsky在[V2]中证明的，我们的证明使用了他的许多技术，但它没有使用他在[V3]中发展的∆闭类理论。1. 预轴上的局部投影模型结构我们首先回顾一些已知的简单预轴上的模型结构。定义1.1。如果f∗:π0(X)→π0(Y)能推导出关联轴的同构，且对于所有U∈T， f∗:πn（X， X）→πn（Y, f(X)）对任意基点X∈X(U)都能推导出关联轴在T/U上的同构，则简化预轴（或多个轴）的映射f: X→Y是一个局部弱等价。如果对于所有U∈T，映射f(U): X(U)→Y (U)是简单集的弱等价（分别为Kan纤维化），则映射f是一个分段弱等价（分别为分段纤维化）。Heller [He]在简单预轴上发现了一种模型结构，其弱等价是分段弱等价。我们将把他的模型结构称为注入模型结构。日期：2001年1月11日。我要感谢Dan Isaksen提出的许多有益的建议，我要感谢我的导师Peter May的鼓励和仔细阅读了许多草稿。我还要感谢Vladimir Voevodsky，因为他注意到了早期版本中的一个错误，也感谢他的工作启发了我写这篇文章。1
We give a model structure on the category of simplicial presheaves on some essentially small Grothendieck site T . When T is the Nisnevich site it specializes to a proper simplicial model category with the same weak equivalences as in [MV], but with fewer cofibrations and consequently more fibrations. This allows a simpler proof of the comparison theorem of [V2], one which makes no use of ∆-closed classes. The purpose of this note is to introduce different model structures on the categories of simplicial presheaves and simplicial sheaves on some essentially small Grothendieck site T and to give some applications of these simplified model categories. In particular, we prove that the stable homotopy categories SH((Sm/k)Nis, A 1) and SH((Sch/k)cdh, A 1) are equivalent. This result was first proven by Voevodsky in [V2] and our proof uses many of his techniques, but it does not use his theory of ∆-closed classes developed in [V3]. 1. The local projective model structure on presheaves We first recall some of the other well-known model structures on simplicial presheaves. Definition 1.1. A map f : X → Y of simplicial presheaves (or sheaves) is a local weak equivalence if f∗ : π0(X) → π0(Y ) induces an isomorphism of associated sheaves and, for all U ∈ T , f∗ : πn(X,x) → πn(Y, f(x)) induces an isomorphism of associated sheaves on T/U for any choice of basepoint x ∈ X(U). The map f is a sectionwise weak equivalence (respectively sectionwise fibration) if for all U ∈ T , the map f(U) : X(U) → Y (U) is a weak equivalence (respectively Kan fibration) of simplicial sets. Heller [He] discovered a model structure on simplicial presheaves whose weak equivalences are the sectionwise weak equivalences. We will refer to his model structure as the injective model structure. Date: January 11, 2001. I would like to thank Dan Isaksen for his many helpful suggestions, and I thank my adviser Peter May for his encouragement and careful reading of many drafts. I am also grateful to Vladimir Voevodsky for noticing an error in an earlier version and for his work that inspired this note. 1